Harmonic Series

$\displaystyle {\begin{alignedat}{8}1&+{\frac {1}{2}}&&+\Big({\frac {1}{3}}&&+{\frac {1}{4}}\Big)&&+\Big({\frac {1}{5}}&&+{\frac {1}{6}}&&+{\frac {1}{7}}&&+{\frac {1}{8}}\Big)&&+\Big({\frac {1}{9}}&&+\cdots \\ {}\geq 1&+{\frac {1}{2}}&&+\Big({\frac {1}{\color {red}{\mathbf {4} }}}&&+{\frac {1}{4}}\Big)&&+\Big({\frac {1}{\color {red}{\mathbf {8} }}}&&+{\frac {1}{\color {red}{\mathbf {8} }}}&&+{\frac {1}{\color {red}{\mathbf {8} }}}&&+{\frac {1}{8}}\Big)&&+\Big({\frac {1}{\color {red}{\mathbf {16} }}}&&+\cdots \\ {}= 1&+{\frac {1}{2}}&& +\ \ {\frac {1}{2}}&& && +\ \ {\frac {1}{2}}&& && && &&+\ \ \ {\frac {1}{2}}&&+\cdots =\infty\end{alignedat}}$ Mercator series $\displaystyle \ln 2 = 1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots$ Leibniz Series for $

High School Physics - Kinematics

Displacement $\vec{x(t)}$, Velocity $\vec{v(t)}$, and Acceleration $\vec{a(t)}$: $\lim\limits_{\Delta t\to 0}\frac{\vec{\Delta x}}{\Delta t} = \vec{v} = \int_{t_i}^{t_f}\vec{a}dt$ Uniform Accelerated Motion ($\vec{a}$ has constant magnitude, direction) $\boxed{5-3-2}$ $\displaystyle \vec{\Delta x} = \vec{v_i}\Delta t+\frac{1}{2}\vec{a}(\Delta t)^2 = \vec{v_f}\Delta t-\frac{1}{2}\vec{a}(\Delta t)^2$ $\displaystyle{\vec{v_f^2} = \vec{v_i^2}+2\vec{a}\vec{\Delta{x}}\Leftarrow \begin{cases}\vec{v}_f = \vec{v}_i+\vec{a}\Delta t\\ \vec{\Delta x} = \Big(\frac{\vec{v_i}+\vec{v_f}}{2}\Big) \Delta t\end{cases}}$


Computer Science Style Guide for Python Code Mathmatics AoPS: Theorems The Mathematics Genealogy Project Understanding Quaternions

Maclaurin Series and Euler's Formula

Maclaurin Series: $\displaystyle f(x)=a_0+a_1x+a_2x^2+\cdots +a_nx^n+\cdots\ \ \ and \ \ a_n=\frac{f^n(0)}{n!}$ $\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots +{\frac {x^{n}}{n!}}+\cdots \quad \forall x$ $\displaystyle \sin x =\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots\forall x $ $\displaystyle \cos x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \forall x$ Euler's Formula: $\displaystyle \boxed{e^{ix}=\cos x +i\sin x}$ $\displaystyle =1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+i\frac{x^5}{5!}+\cdots=\Big(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots\Big)+i\Big(x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots\Big)$



The Benefits of Globalization

A Grade 10 position paper against the following quote. "Globalization, as defined by rich people like us, is a very nice are talking about the internet, you are talking about cell phones, you are talking about computers.  This doesn't affect two-thirds of the people in the world."  - Jimmy Carter The Benefits of Globalization Ever since the Industrial Revolution, globalization has made goods cheaper and more accessible to the general population than ever before.  However, one such critic in Jimmy Carter states that "rich people," in which he is referencing citizens of developed countries such as the United States, are the main benefactors from globalization, because they have access to resources such as "the Internet,...cell phones,...[and] computers."  He also states that globalization "doesn't affect two-thirds of the people of the world," referring to citizens of developing countries.  It is also worth noting that Carter wa

Memories for the First City Championship - free from striving, no one could strive with

At the beginning of my 9th grade volleyball season, it was very common to find our team predicted to be a mediocre team at best.  After all, we lacked consistent and strong hitters, height, and team chemistry at the time.  Unfortunately, through most of the season this held true.  However, when "clutch time" came, we shone, and ultimately, having no expectations became the difference maker. The regular season went by extremely fast.   We were ignored and looked down upon during the entire season, and finished as a lousy 5th in the standings.  Our chances at a title were slim, and it looked like another dismal season for Vernon Barford Blues. Soon after, playoffs came along.  There are no participation awards in tier one; only gold, silver, or bronze.  Our coach made us prepare for the upcoming storm by having practice almost every day.  We did not want to go out without a fight, so nobody complained or talked about nonsense, and practices  became much more focused and inten